IFToMM WC 2019: Advances in Mechanism and Machine Science pp 387-396

# A Novel Dual-Matrix Method for Displacement Analysis of Spatial Linkages

• Yu Zhang
• Song Lin
• Jingyu Jiang
Conference paper
Part of the Mechanisms and Machine Science book series (Mechan. Machine Science, volume 73)

## Abstract

In view of the limitation of general dual algebra approach to kinematic analysis, this paper proposes a novel dual-matrix method as a unified procedure for displacement analysis of spatial linkages. By this mean, the Euler angles computation of spherical joints is avoided, and the coordinate transformation equation of spatial linkages which contain varied kinds of joints can be solved more easily and efficiently. Firstly, the transformation equation of spatial linkages is established by dual-matrix method. Secondly, according to the positional relation of the spherical joint and linkages, the rotation transformation matrix containing Euler angles in the equation is eliminated. Finally, the spatial linkages kinematic equation is obtained after rearranging and simplifying matrix equation. This paper takes RSSR mechanism as an example to illustrate the improved dual-matrix approach, and kinematic formulas are derived. The kinematic simulation is performed in computer and then compared with the results of formula calculation, which proves that this method is completely correct.

## Keywords

Spatial Linkages Dual-Matrix Displacement Analysis RSSR Mechanism

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## References

1. 1.
Duffy, J., Gilmartin, M. J.: Displacement analysis of the generalised RSSR mechanism. Mechanism & Machine Theory, 13(5), pp. 533-541(1978).Google Scholar
2. 2.
Hiller, M., Woernle, C.: A unified representation of spatial displacements. Mechanism & Machine Theory, 19(6), pp. 477-486(1984).Google Scholar
3. 3.
Yang A.T.: Displacement analysis of spatial five-link mechanisms using (3×3) matrices with dual-number elements. ASME. Journal of Engineering for Industry, 91(1), pp. 152-156(1985).
4. 4.
Pennock, G. R., Yang, A. T.: Application of dual-number matrices to the inverse kinemat- ics problem of robot manipulators. Journal of Mechanical Design, 107(2), pp. 201-208(1985).
5. 5.
Bai, S., Angeles, J.: A robust solution of the spatial Burmester problem. Journal of Mecha- nisms & Robotics, 4(3), 031003(2012).Google Scholar
6. 6.
Venkataramanujam, V., Larochelle, P. M.: A coordinate frame useful for rigid-body dis- placement metrics. Journal of Mechanisms & Robotics, 2(4), pp. 317-337(2010).Google Scholar
7. 7.
Wang, G., Liu, X.: Role and influence of modern mathematics in mechanisms. Journal of Mechanical Engineering, 49(3), pp. 1-9(2013).
8. 8.
McCarthy, J. M., Soh, G. S.: Geometric design of linkages. Springer-Verlag(2011).Google Scholar
9. 9.
Liao, Q.: Geometry algebra method for solving the kinematics of linkage mechanisms. Journal of Beijing University of Posts and Telecommunications, 33(4), pp. 1-11(2010).Google Scholar
10. 10.
A.G. Erdman: Modern Kinematics Developments in the Last Forty Years. Wiley, New York, pp. 451–463(1993).Google Scholar
11. 11.
Fischer, I.: Dual-number methods in kinematics, statics and dynamics. Crc Press(1999).Google Scholar
12. 12.
Pandrea, N., Popa, D., Stanescu, N. D.: Classical and Modern Approaches in the Theory of Mechanisms. Wiley, Hoboken, N. J., pp. 317–320(2017).